The global existence and the finite time blow-up for the problem are proved by the potential well method at both low and critical initial energy levels

Bull. Korean Math. Soc. 55 (2018), No. 4, pp. 1161–1178 https://doi.org/10.4134/BKMS.b170634

pISSN: 1015-8634 / eISSN: 2234-3016

GLOBAL SOLUTIONS FOR A CLASS OF NONLINEAR SIXTH-ORDER WAVE EQUATION

Ying Wang

Abstract. In this paper, we consider the Cauchy problem for a class of nonlinear sixth-order wave equation. The global existence and the finite time blow-up for the problem are proved by the potential well method at both low and critical initial energy levels. Furthermore, we present some sufficient conditions on initial data such that the weak solution exists globally at supercritical initial energy level by introducing a new stable set.

1. Introduction

This paper is concerned with the Cauchy problem for the following nonlinear sixth-order wave equation

utt− uxxtt− uxx+ uxxxx+ uxxxxtt= f (ux)x, (1.1)

u(x, 0) = φ(x), u_t(x, 0) = ψ(x), (1.2)

where x ∈ R, u(x, t) is the unknown function, f (s) is a given nonlinear function.

In order to investigate the water wave problem with surface tension, Schnei- der and Wayne [12] considered a class of Boussinesq equation which models the water wave problem with surface tension as follows

utt= uxx+ uxxtt+ µuxxxx− uxxxxtt+ (u²)xx, (1.3)

where x, t, µ ∈ R and u(t, x) ∈ R. The model can also be formally derived from the 2D water wave problem. Eq. (1.3) with µ > 0 is known as the “good”

Boussinesq equation because of its linear instability. For a degenerate case, they have proved that the long wave limit can be described approximately by two decoupled Kawahara-equations. The authors included the term with the sixth-order derivatives since they were interested precisely in the case when the coefficient in front of the term with the fourth-order derivative is small, i.e., 1 + µ = ε²ν, with ν ∈ R fixed [12]. The lowest order nonlinear terms in the water wave problem remain unchanged from the classical equation because they

Received July 19, 2017; Revised December 2, 2017; Accepted December 29, 2017.

2010 Mathematics Subject Classification. 35L60, 35K55, 35Q80.

Key words and phrases. Cauchy problem, sixth-order wave equation, blow-up, existence of global solution.

c

2018 Korean Mathematical Society 1161

are independent of surface tension. Wang [20, 21] studied the well-posedness of the local and global solution, the blow-up of solutions and nonlinear scattering for small amplitude solutions to Eq. (1.3) in R and Rⁿ. The global existence and finite time blow-up of the solutions for Eq. (1.3) are established by the potential well method [17]. Eq. (1.1) is similar to the generalized Boussinesq equation and various generalized of the Boussinesq equations have been studied from many aspects [5, 6, 8–10, 13–15].

Wang and Xu [16] considered the Cauchy problem for the following Rosenau equation

utt+ uxxxx+ uxxxxtt− ruxx= f (u)xx. (1.4)

Under some conditions, the well-posedness of the solution and the nonexistence of global solution to the problem are proved with the aid of the potential well method. The authors [23] studied the following nonlinear wave equation

utt+ uxxxx+ uxxxxtt− ruxx= φ(ux)x. (1.5)

The existence and nonexistence of global solutions to Eq. (1.5) are obtained by the potential well method. Most of the authors proved the global existence and finite time blow-up of the solution at the sub-critical initial energy level (E(0) < d) and critical initial energy level (E(0) = d) by the potential well- method (the definitions of E(t) and d will be given later). In particular, to our knowledge, there have only been a few results up to now on the global existence of a solution to the Cauchy problem (1.1) and (1.2) at the high initial energy level (E(0) > 0).

Hatice et al. [18, 19] considered the following Rosenau equation (1.6) utt− uxx+ uxxxx+ uxxxxtt= (f (u))xx, x ∈ R, t > 0,

where f (u) = γ|u|^p, γ > 0. By defining new functionals and using potential well method, they established the existence of global weak solutions for Eq. (1.6) with supercritical initial energy (E(0) > 0). Furthermore, the authors general- ized the results from one dimension to n-dimensional spaces. Then, Kutev et al. [4] considered the Cauchy problem for the Boussinesq paradigm equation (1.7) utt− ∆u − β1∆utt+ β2∆²u = ∆f (u), (x, t) ∈ Rⁿ× (0, +∞), where f (u) = α|u|^p, α > 0, β1≥ 0, β2> 0, and ∆ denotes the Laplace operator in Rⁿ. They introduced some new functionals and gave the existence of global weak solution with supercritical initial energy E(0) > 0. But the method in [4, 18, 19] is not valid for the nonlinear term f (u) = β|u|^pu, β < 0 at arbitrarily positive initial energy level E(0) > 0.

Wang and Mu [21] considered the following Boussinesq equation (1.8) utt− ∆utt+ ∆²utt− ∆u + ∆²u = ∆f (u), (x, t) ∈ Rⁿ× (0, +∞).

They obtained the existence and the uniqueness of the global solution and blow- up of the solution under some restrictions on the nonlinear source term f (u).

When f (u) = ±β|u|^por −β|u|^p−1u, β > 0, Xu et al. [24] considered the Cauchy

problem of Eq. (1.8) at three different initial energy levels. They established the global existence and blow-up solutions at low and critical initial energy levels, and also proved the global existence of the weak solution at supercritical initial energy level. In this paper, we apply the potential well method ([7, 22, 24]) to study the global existence and nonexistence of the solution to the problem (1.1) and (1.2) at the sub-critical initial energy level, the critical initial energy level and the high initial energy level.

In the present paper, using the contraction mapping principle, the well- posedness for problem (1.1) and (1.2) was established in Section 2. Under some conditions of f (u), we also prove the existence and uniqueness of the global solution for the problem (1.1) and (1.2) in Section 2. In sections 3 and 4, we fix the nonlinear terms to be f (u) = r|u|^p, r 6= 0 to classify the initial data for the global existence and non-global existence. Following the main idea of potential well method introduced in [7, 22, 24], we will construct the stable and unstable sets and study the global existence and non-existence of the weak solutions for problem (1.1) and (1.2) at three different initial energy levels:

E(0) < d, E(0) = d and E(0) > 0, where d is the potential well depth, and the three cases will be respectively dealt with different methods.

Throughout this paper, L^p denotes the usual space of all L^p(R)-functions with norm k · kp and kuk = kuk2, H^s denotes the Sobolev space H^s(R) with norm k·kH^sand H₀¹(R) denotes the closure of Cc^∞(R) in H¹(R). If not specified throughout this paper, we denote C as a generic constant varying line by line and depending only on the norms of initial data and absolute constant.

At first, by using the contraction mapping theorem, we obtain the following existence and uniqueness of the local solution to problem (1.1) and (1.2).

Theorem 1.1. Suppose that s > ¹₂, φ, ψ ∈ H^s and f ∈ C^N(R), then there exists a maximal time T0which depends on φ and ψ such that for each T < T0, the Cauchy problem (1.1) and (1.2) has a unique solution u ∈ C¹([0, T ], H^s).

Moreover, if

sup

t∈[0,T0)

[ku(·, t)k_Hs+ ku_t(·, t)k_Hs] < ∞, then T0= ∞.

Theorem 1.2. Suppose that s ≥ 1, φ, ψ ∈ H^s, F (u) =Ru

0 f (z)dz, F (φ_x) ∈ L¹ and T₀ > 0 is the maximal existence time of corresponding solution u(t) ∈ C¹([0, T₀), H^s) to the Cauchy problem (1.1) and (1.2). Then the equality

E(t) = 1

2[ku_tk²+ ku_xtk²+ ku_xk²+ ku_xxk²+ ku_xxtk²] + Z

R

F (u_x)dx

= E(0), ∀t ∈ (0, T0), (1.9)

holds.

Under some assumptions, the well-posedness of the global solution for the Cauchy problem (1.1) and (1.2) is established.

Theorem 1.3. Suppose that the assumptions of Theorem 1.1 hold, and T₀> 0 is the maximal existence time of the corresponding solution u(t) ∈ C²([0, T0], H^s) to the problem (1.1) and (1.2). Then T0< ∞ if and only if

lim

t→T₀sup ku_x(t)k_L^∞ = ∞.

For the case f (s) = r|u|^p, r 6= 0, we firstly introduce the potential energy functional

(1.10) J (u) = 1

2(kuxk²+ kuxxk²) + r p + 1

Z

R

|ux|^puxdx and the Nehari functional

(1.11) I(u) = (kuxk²+ kuxxk²) + r Z

R

|ux|^puxdx.

We define the stable set

(1.12) K₁= {u ∈ H²| I(u) > 0} ∪ {0}, the unstable set

(1.13) K2= {u ∈ H²| I(u) < 0}

and the depth of potential well as d = inf

u∈N EJ (u),

where the Nehari manifold N E = {u ∈ H²\ {0} | I(u) = 0}. And for the function u(x, t) satisfying u ∈ C¹((0, T ), H²), ut ∈ C((0, T ), H²), we define a new functional space

YT := {u|I(u(t)) > kutk²+ kuxtk²+ kuxxtk²} ∪ {0}, which will be used in Section 4.

For the low initial energy case and the critical energy case, we prove the global existence and finite-time blow-up for the problem (1.1) and (1.2) by the potential well method.

Theorem 1.4. Suppose that 2 ≤ s < p + 1 and φ, ψ ∈ H^s. If E(0) ≤ d and φ ∈ K1∪ ∂K1, then problem (1.1) and (1.2) has a unique solution u ∈ C¹([0, ∞); H^s) and u(t) ∈ K1∪ ∂K1 for t ∈ [0, ∞).

Theorem 1.5. Suppose that 2 ≤ s < p+1 and φ, ψ ∈ H^s. If E(0) < d, φ ∈ K2

and (φ, ψ) + (φx, ψx) + (φxx, ψxx) ≥ 0 when E(0) = d, then the solution u(x, t) of problem (1.1) and (1.2) ceases to exist in finite time.

For the supercritical initial energy case, by utilizing the method of [4,24], we obtain the global existence of weak solution for problem (1.1) and (1.2) under some sufficient conditions on the initial data.

Theorem 1.6. Suppose that 2 ≤ s < p + 1 and φ ∈ H²∩ H₀¹, ψ ∈ H²∩ H₀¹. If E(0) > 0,

(1.14)

I(φ) > kψk²+ kψxk²+ kψxxk², (1.15)

2(φ, ψ) + 2(φx, ψx) + 2(φxx, ψxx)

+ kφk²+ kφ_xk²+ kφ_xxk²+2(p + 1)

p + 3 E(0) < 0, (1.16)

then the solution u(x, t) of problem (1.1) and (1.2) exists globally.

2. Well-posedness of solutions

In this section, we study the local and global well-posedness of solutions to the problem (1.1) and (1.2). We first consider the following linear wave equation

u_tt− u_xxtt− u_xx+ u_xxxx+ u_xxxxtt= q(x, t), (2.1)

with initial value (1.2). By the Fourier transform and Duhamel’s principle, the solution u(x, t) of problem (2.1) and (1.2) can be written as

(2.2) u(x, t) = (∂_tS(t)φ)(x) + (S(t)ψ)(x) + Z t

0

Γ(t − τ )q(u(τ ))dτ.

Here Γ(t) = S(t)(1 − ∂²_x+ ∂_x⁴)⁻¹ and (∂tS(t)φ)(x) = 1

2π Z

R

e^ixξcos( |ξ|p 1 + ξ²t

p1 + ξ²+ ξ⁴) ˆφ(ξ))dξ, (S(t)ψ)(x) = 1

2π Z

R

e^ixξsin( |ξ|p 1 + ξ²t p1 + ξ²+ ξ⁴)

p1 + ξ²+ ξ⁴

|ξ|p 1 + ξ²

ψ(ξ))dξ,ˆ

where φ(ξ) = F (φ)(ξ) =ˆ R

Re^−ixξφ(x)dx is the Fourier transform of φ(x).

Lemma 2.1. For the operators ∂_tS(t), S(t) and Γ(t), we have the following estimates

k∂tS(t)φkH^s ≤ kφkH^s, ∀φ ∈ H^s, (2.3)

kS(t)ψkH^s≤ 2(1 + t)kψkH^s, ∀ψ ∈ H^s, (2.4)

k∂ttS(t)φk_Hs−2 ≤√

2kφkH^s, ∀φ ∈ H^s, (2.5)

kΓ(t)qkH^s ≤√

2kqk_Hs−4, ∀q ∈ H^s−4, (2.6)

k∂tΓ(t)qk_Hs−2 ≤ kqk_Hs−4, ∀q ∈ H^s−4. (2.7)

Proof. We only prove (2.4) and (2.6), the proof of other inequalities are similar.

By Plancherel’s theorem, we obtain kS(t)ψk²_Hs =

Z

R

(1 + ξ²)^s1 + ξ²+ ξ⁴

ξ²(1 + ξ²) sin²( t|ξ|p 1 + ξ²

p1 + ξ²+ ξ⁴)| ˆψ(ξ)|²dξ

≤ Z

|ξ|≤1

(1 + ξ²)^st²| ˆψ|²dξ + Z

|ξ|>1

(1 + ξ²)^s(1 + ξ²+ ξ⁴)

ξ²(1 + ξ²) | ˆψ(ξ)|²dξ

≤ t² Z

|ξ|≤1

(1 + ξ²)^s| ˆψ(ξ)|²dξ + 4 Z

|ξ|>1

(1 + ξ²)^s| ˆψ(ξ)|²dξ

≤ 4(1 + t)²kψk²_Hs,

kΓ(t)qk²_Hs= Z

R

(1 + ξ²)^ssin²(^t|ξ|

√

1+ξ²

√

1+ξ²+ξ⁴)1 + ξ²+ ξ⁴ ξ²(1 + ξ²)

1

(1 + ξ²+ ξ⁴)²|ˆq(ξ)|²dξ

≤ 2 Z

R

(1 + ξ²)^s−4|ˆq(ξ)|²dξ = 2kqk²_Hs−4.

Therefore (2.4) and (2.6) hold. This completes the proof of the lemma.  Lemma 2.2 ([1]). Suppose that g(u) ∈ C^N(R) is a function vanishing at zero, where N ≥ 0 is an integer. Then for any s with 0 ≤ s ≤ N and any u, v ∈ H^s∩ L^∞, we have

kg(u)kH^s ≤ G(kukL^∞)kukH^s,

kg(u) − g(v)k_Hs ≤ ¯G(kuk_L^∞, kvk_L^∞)ku − vk_Hs,

where G : [0, ∞) → R and ¯G : [0, ∞) × [0, ∞) → R are continuous functions.

Lemma 2.3 (Sobolev Lemma [2]). If s > ⁿ₂ + k, where k is a nonnegative integer, then

H^s(Rⁿ) ⊂ C^k(Rⁿ) ∩ L^∞, where the inclusion is continuous. In fact,

Σ_|β|≤kk∂^β_xukL^∞ ≤ CkukH^s, where C is dependent on s, independent of u.

Lemma 2.4 ([11]). Assume u ∈ H^s∩ L^∞, 0 < s < p. Then there exists a constant C such that

k|u|^pk_Hs ≤ Ckuk^p−1_L∞kuk_Hs.

Lemma 2.5 ([3]). If s > 0, then H^s∩ L^∞ is an algebra. Moreover, kuvkH^s ≤ C(kukL^∞kvkH^s+ kvkL^∞kukH^s).

Lemma 2.6. The operator P = (1 − ∂²_x+ ∂_x⁴)⁻¹ is bounded from H^s−4to H^s, i.e.,

kP ukH^s ≤ CkukH^s−4. Proof.

kP ukH^s = k(1 + ξ²)^s2 1 1 + ξ²+ ξ⁴ukˆ

≤ Ck(1 + ξ²)^s−42 uk = Ckukˆ _Hs−4. 

Proof of Theorem 1.1. Now we are going to prove the existence and uniqueness of local solutions for problem (1.1) and (1.2) by contraction mapping argumen- tation. For this purpose, we define the function space X(T ) = C¹([0, T ]; H^s) with s > ¹₂, equipped with the norm defined by

kukX(T )= max

t∈[0,T ]

[ku(·, t)k_Hs+ ku_t(·, t)k_Hs].

Since H^s,→ L^∞ for s > ¹₂, we have u ∈ L^∞ if u ∈ X(T ). Let B_R(T ) be the ball of radius R centered at the origin in X(T ), i.e.,

BR(T ) = {u ∈ X(T ) : kuk_{X(T )}≤ R}.

For φ ∈ H^s, ψ ∈ H^s, u ∈ X(T ), we define the map (2.8) Θ(u(t)) = ∂tS(t)φ + S(t)ψ +

Z t 0

S(t − τ )(I − ∂²_x+ ∂_x⁴)⁻¹f (ux)x(τ )dτ.

It will be shown that Θ : BR(T ) → BR(T ) is contractive if R and T are well chosen.

Let kφkH^s + kψkH^s ≤ ρ, u, v ∈ BR(T ). Using the estimates in Lemmas 2.1-2.5, we have

kΘ(u)kH^s ≤ kφkH^s+ 2(1 + T )kψk_Hs−2+√ 2

Z T 0

kf (ux))k_Hs−3dτ

≤ 2ρ + (2ρ +√

2G(R))RT (2.9)

and (2.10)

kΘ(u) − Θ(v)kH^s≤√ 2

Z T 0

kf (ux)x− f (vx)xkH^sdτ ≤√

2 ˜GT ku − vkX(T ), where ˜G(R) = ¯G(R, R). Differentiating with respect to t, we see that (2.11) ut(x, t) = (∂ttS(t)φ)(x) + (∂tS(t)ψ)(x) +

Z T 0

∂tΘ(t − τ )f (ux)xdτ.

Using the estimates in Lemmas 2.1 and 2.2, we have kΘ(u)_tk_Hs ≤ 2kφk_Hs+ kψk_Hs−2+

Z T 0

kf (u_x)_xk_Hs−4dτ

≤ 2(ρ + G(R)RT ) (2.12)

and

kΘ(u)_t− Θ(v)_tk_Hs ≤ Z T

0

kf (u_x)_x− f (v_x)_xk_Hs−4dτ

≤ G(R)T ku − vk_{X(T )}. (2.13)

From (2.9)-(2.13), we have

kΘ(u)k_{X(T )}≤ 4ρ + (2ρ + 2√

3G(R)R)T, kΘ(u) − Θ(v)k_{X(T )}≤ 3 ¯G(R)T ku − vk_{X(T )}.

Choosing R = 8ρ and fixing T so small enough such that T < min

2ρ ρ+√

3G(R)R,_{4 ¯G(R)}¹  , (2.14)

therefore, Θ is a contraction map on BR(T ). It follows from the contrac- tion mapping theorem that problem (2.1) and (1.2) has a unique solution u ∈ BR(T ). Similar to that of [20], we can prove uniqueness and local Lipschitz dependence with respect to the initial data in the space BR(T ). Using unique- ness we can extend the result in the space C([0, T0]; H^s(R)) by a standard

technique. 

Proof of Theorem 1.2. It follows from (1.1) that d

dtE(t) = (utt, ut) + (ux, uxt)

+ (u_xx, u_xxt) + (u_xtt, u_xt) + (u_xxtt, u_xxt) + (f (u_x), u_xt)

= h(utt− uxxtt+ uxxxxtt− uxx+ uxxxx− f (ux)x, utiX∗X= 0, where (·, ·) denotes the inner product of L²-space and h·, ·iX∗Xmeans the usual duality of X and X with X = H². Integrating the above equality with respect

to t, we have (1.9). Theorem 1.2 is proved. 

Next we study the existence of global solutions to the problem (1.1) and (1.2).

Proof of Theorem 1.3. One implication is obvious in view of Theorem 1.1. Let us prove that if

(2.15) lim

t∈[0,T0)

sup kux(·, t)kL^∞ = M < ∞, then T0= ∞. (1.1) can be written as follows:

utt+ u = P u + P f (ux)x, where P = (1 − ∂_x²+ ∂_x⁴)⁻¹.

So, it follows from H¨older inequality that for t ∈ (0, T ), 1

2 d

dt(kuk²_Hs+ ku_tk²_Hs)

= ((I − ∂_x²)^s2utt, (I − ∂²_x)^s2ut) + ((I − ∂_x²)^s2u, (I − ∂²_x)^s2ut)

= ((I − ∂_x²)^sutt+ (I − ∂_x²)^su, ut)

= ((I − ∂_x²)^s(utt+ u), ut)

= ((I − ∂_x²)^s2(P u + P (∂xf (ux))), (I − ∂_x²)^s2ut)

≤ k(I − ∂_x²)^s2(P u + P (∂xf (ux)))kH^skutkH^s

≤ (k(I − ∂²_x)^s2P ukH^s+ k(I − ∂²_x)^s2P (∂xf (ux))kH^s)kutkH^s. From Lemma 2.4, Lemma 2.6 and (3.1), we obtain

(k(I − ∂_x²)^s2P ukH^s+ k(I − ∂_x²)^s2P (∂xf (ux))kH^s) ≤ (G(R) + C)kukH^s.

It follows from the Cauchy inequality that 1

2 d

dt(kuk²_Hs+ kutk²_Hs) ≤ (G(R)²+ C)(kuk²_Hs+ kutk²_Hs), t ∈ (0, T ).

Then previous relation shows by Gronwall’ inequality that ku(t)k²_Hs+kut(t)k²_Hs

do not blow-up in finite time. Then T = ∞ by Theorem 1.1. The theorem is

proved. 

3. Existence and nonexistence of global solutions for f (u) = r|u|^p In this section, we discuss the existence and nonexistence of global solution for the low initial energy case and the critical energy case when f (s) = r|u|^p, r 6=

0. We start with the following elementary statement.

Lemma 3.1. The depth of potential well d = _2(p+1)^p−1 (|r|S_∗^p+1)^−p−12 , where S_∗ is the optimal Sobolev constant, i.e.,

(3.1) S_∗= sup

u∈H²\{0}

kuxkp+1

(ku_xk²+ ku_xxk²)¹².

Proof. From the definition of d, we obtain u ∈ N E, i.e., I(u) = 0, which yields kuxk²+ ku_xxk²= −rku_xk^p+1_p+1 ≤ |r|S_∗^p+1(ku_xk²+ ku_xxk²)^p−12 (ku_xk²+ ku_xxk²).

From (3.1), we get

kuxk²+ kuxxk²≥ (|r|S_∗^p+1)^−p−12 . (3.2)

On the other hand, from (1.11), (1.12), (3.2) and I(u) = 0, we obtain J (u) = (1

2 − 1

p + 1)(kuxk²+ kuxxk²)

≥ p − 1

2(p + 1)(|r|S_∗^p+1)^−p−12 , (3.3)

which completes the proof. 

Lemma 3.2. Let φ, ψ ∈ H², and u ∈ C¹([0, T₀); H²) is the unique solution of the Cauchy problem (1.1) and (1.2), where T₀is the maximal existence time of u(t). Assume that E(0) < d, then for all t ∈ [0, T₀),

(i) u(t) ∈ K1 and kux(t)k²+ kuxx(t)k²<^2(p+1)_p−1 d if φ ∈ K1; (ii) u(t) ∈ K₂ and ku_x(t)k²+ ku_xx(t)k²> ^2(p+1)_p−1 d if φ ∈ K₂.

Proof. Since the proof of (i) and (ii) are similar, we only prove (ii). Let u(t) be any local weak solution of problem (1.1) and (1.2) with E(0) < d, φ ∈ K2and T0

be the maximum existence time of u(t). Then, it follows from Theorem 1.2 that E(u(t)) = E(0) < d. Thus, it suffices to show that I(u(t)) < 0 for 0 < t < T0. Arguing by contradiction, we suppose that there exists a t1∈ (0, T0) such that

I(u(t₁)) ≥ 0. From the continuity of I(u(t)) in time, there exists a t_∗∈ (0, T₀) such that I(u(t_∗)) = 0. Then, from the definition of d, we get

d ≤ J (u(t∗)) ≤ E(u(t∗)) = E(0) < d,

which is a contradiction, that is ku_xk²+ku_xxk²< −rku_xk^p+1_p+1for any t ∈ [0, T₀).

From (3.1), we arrive at 1

S_∗² ≤kuxk²+ kuxxk²

kuxk²_p+1 <−rkuxk^p+1_p+1

kuxk²_p+1 ≤ |r|kuxk^p−1_p+1. (3.4)

Then, from Lemma 3.1 and (3.4), yields d = p − 1

2(p + 1)|r|^−p−12 S_∗⁻²S⁻

4

∗ p−1

< p − 1 2(p + 1)

kuxk²+ kuxxk²

kuxk²_p+1 (ku_xk^p−1_p+1)^p−12

= p − 1

2(p + 1)(ku_xk²+ ku_xxk²), which means

kuxk²+ kuxxk²> 2(p + 1) p − 1 d.

In fact, for (i), from the definition of I(t) and E(t), we get E(t) = 1

2[kutk²+ kuxtk²+ kuxk²+ kuxxk²+ kuxxtk²] + r p + 1

Z

R

|ux|^puxdx

= 1

2(kutk²+ kuxtk²+ kuxxtk²) + p − 1

2(p + 1)(kuxk²+ kuxxk²) + 1 p + 1I(t).

If I(u(t)) > 0, we obtain p − 1

2(p + 1)(kuxk²+ kuxxk²) < E(t) = E(0) < d.

So,

kuxk²+ kuxxk²< 2(p + 1) p − 1 d.

This completes the proof of Lemma 3.2. 

Lemma 3.3. Let φ, ψ ∈ H², and u ∈ C¹([0, T0); H²) is the unique solution of the Cauchy problem (1.1) and (1.2), where T0is the maximal existence time of u(t). Assume that E(0) = d and (φ_xx, ψ_xx) + (φ_x, ψ_x) + (φ, ψ) ≥ 0, then for all t ∈ [0, T₀), u(t) ∈ K₂ if u(0) ∈ K₂.

Proof. If the result is false, there would exist a t₀∈ (0, T0) such that I(t₀) = 0 from the continuity of I(t). Hence we have J (u(t₀)) ≥ d, which together with E(t₀) = E(0) = d gives J (u(t₀)) = d and

kut(t0)k + kuxt(t0)k + kuxxt(t0)k = 0.

(3.5)

On the other hand, let

L(t) = ku(t)k²+ kux(t)k²+ kuxx(t)k². (3.6)

Then

L⁰(t) = 2(u, ut) + 2(ux, uxt) + 2(uxx, uxxt), (3.7)

with L⁰(0) ≥ 0. From (1.1) and (1.8), we get

L⁰⁰(t) = 2hutt− uxxtt+ uxxxxtt, uiX∗X+ 2kutk²+ 2kuxtk²+ 2kuxxtk²

= −2kuxk²−2kuxxk²−2r Z

R

|ux|^puxdx + 2kutk²+ 2kuxtk²+ 2kuxxtk²

= 2kutk²+ 2kuxtk²+ 2kuxxtk²− 2I(u)

> 0, ∀t ∈ (0, t₀).

(3.8)

Hence L⁰(t) is strictly increasing on [0, t0], and L⁰(t0) > 0 which contradicts

(3.5). The Lemma is proved. 

Lemma 3.4. Let φ ∈ H^s, ψ ∈ H^s, 2 ≤ s ≤ p + 1. Assume that E(0) < d, then when φ ∈ K1, problem (1.1) and (1.2) has a unique solution u ∈ C¹([0, ∞); H^s) and u ∈ K1 for 0 ≤ t < ∞.

Proof. From Lemma 3.2, we can obtain

kux(t)k²+ ku_xx(t)k²< 2(p + 1)d p − 1 .

It follows from above inequality and the Sobolev imbedding theorem that sup

t∈[0,T0)

kux(·, t)k²_L∞ < C2(p + 1)d p − 1 .

Therefore, problem (1.1) and (1.2) has a unique global solution u ∈ C¹([0, ∞);

H^s) by Theorem 1.3. 

Proof of Theorem 1.4. First, it follows from Theorem 1.3 that problem (1.1) and (1.2) has a unique local solution u ∈ C¹([0, T₀), H^s), where T₀ is the maximal existence time of u(t). Next we prove T0= ∞.

It follows from φ ∈ K1∪ ∂K1, E(0) ≤ d and the proof of Lemma 3.2 that kφxk^p+1_p+1≤ S_∗^p+1(kφxk²+ kφxxk²)^p+12

≤ S_∗^p+1(kφ_xk²+ kφ_xxk²)(2(p + 1)d p − 1 )^p−12

= |r|⁻¹(kφ_xk²+ kφ_xxk²).

Therefore for any λ > 0, we obtain J (λu) = λ²

2 [ku_xk²+ ku_xxk²] +rλ^p+1 p + 1

Z

R

|u_x|^pu_xdx

and

d

dλJ (λφ) = λ(kφxk²+ kφxxk²) + rλ^p Z

R

|φx|^pφxdx

≥ λ(kφxk²+ kφxxk²) − |r|λ^pkφxk^p+1_p+1

≥ λ(1 − λ)(kφxk²+ kφxxk²) ≥ 0, ∀λ ∈ (0, 1).

Take a sequence {λ_m} such that 0 < λ_m < 1, m = 1, 2, . . . and λ_m → 1 as m → ∞. Let φ_m = λ_mφ, ψ_m = λ_mψ. Consider the problem (1.1) with the initial conditions

u(x, 0) = φ_m(x), u_t(x, 0) = ψ_m(x).

(3.9) Then

kφmxk²+ kφmxxk²= λ²_m[kφmxk²+ kφmxxk²] < 2(p + 1)d p − 1 and

Em(0) = 1

2[kψmk²+ kψmxk²+ kψmxxk² + kφ_mxk²+ kφ_mxxk²] + r

p + 1 Z

R

|φmx|^pφ_mxdx,

= 1

2[kψmk²+ kψmxk²+ kψmxxk²] + J (λmφ).

If ψ = 0 and φ = 0, then Em(0) = 0 < d. If ψ 6= 0 and φ 6= 0, then E_m(0) < 1

2[kψ_mk²+ kψ_mxk²+ kψ_mxxk²] + J (φ) = E(0) ≤ d.

It follows from Lemma 3.4 that for each m, problem (1.1) and (3.9) has a unique global solution um∈ C²([0, ∞); H^s) and it satisfies

(umt, v) + (umxt, vx) + (umxxt, vxx) +

Z t 0

[(umx, vx) + (umxx, vxx) + (r|umx|^p, vx)]dτ

= (ψm, v) + (ψmx, vx) + (ψmxx, vx), ∀v ∈ H², t ∈ [0, ∞) (3.10)

and

1

2[kumtk²+ kumxtk²+ |umxxtk²] + J (um) = Em(0) < d, (3.11)

I(um) > 0, (3.12)

kumxk²+ kumxxk²< 2(p + 1) p − 1 d.

(3.13)

By using (3.11)-(3.13), we get

kumxk^p+1_p+1≤ S_∗^p+1(kumxk²+ kumxxk²)^p+12 ≤ |r|⁻¹(kumxk²+ kumxxk²), J (u_m) ≥ 1

2[ku_mxk²+ ku_mxxk²] − |r|

p + 1 Z

R

|u_mx|^p+1dx

≥ p − 1

2(p + 1)[kumxk²+ kumxxk²] ≥ 0.

Because of the above inequalities, we have

kumtk²+ kumxtk²+ kumxxtk²≤ 2d, (3.14)

kumk^p+1_p+1≤ |r|⁻¹2(p + 1) p − 1 d.

(3.15)

It follows from (3.11), (3.12), (3.14) and (3.15) that there exist a ¯u ∈ ¯K₁ and a subsequence {u_k} such that as k → ∞, ukx→ ¯u_xin L^∞(0, ∞; H¹) weakly star and a.e in R × [0, ∞), ukt→ ¯u_t in L^∞(0, ∞; L²) weakly star, |u_kx|^p→ | ¯u_x|^pin L^∞(0, ∞; L^p+1p ) weakly star.

In equality (3.10), letting m = k → ∞, we obtain (¯ut, v) + (¯uxt, vx) + (¯uxxt, vxx) +

Z t 0

[(¯ux, vx) + ¯uxx, vxx) + (r|¯ux|^p, vx)]dτ

= (ψ, v) + (ψx, vx) + (ψxx, vxx), ∀t ∈ [0, ∞),

for any v ∈ H², which implies that ¯u satisfies (1.1). Furthermore, we can get

¯

u(x, 0) = φ(x), ¯ut(x, 0) = ψ(x).

Thus ¯u is a global solution of the Cauchy problem (1.1) and (1.2). From the uniqueness of the solution of problem (1.1) and (1.2), we get ¯u = u on R×[0, T⁰), and I(u) = I(¯u) ≥ 0. We obtain T0= ∞ and u ∈ C²([0, ∞); H^s). The theorem

is proved. 

Proof of Theorem 1.5. Theorem 1.1 gives the existence of a local weak solution u ∈ C¹([0, T0); H^s) satisfying (1.9), where T0 is the maximal existence time of u. We prove T₀< ∞. We assume that the result is not true, then T₀= ∞. Let

L(t) = ku(t)k²+ kuxk²+ kuxx(t)k², (3.16)

then

L⁰(t) = 2(u, ut) + 2(ux, uxt) + 2(uxx, uxxt).

Using the Schwartz inequality, we have

L⁰(t)²≤ 4L(t)[kutk²+ kuxtk²+ kuxxtk²].

(3.17)

Similarly to (3.8), from Lemmas 3.2 and 3.3, we obtain

(3.18)

L⁰⁰(t) = (p + 3)[kutk²+ kuxtk²+ kuxxtk²] + (p − 1)[kuxk²+ kuxxk²]

− 2(p + 1)E(0)

> (p + 3)[ku_tk²+ ku_xtk²+ ku_xxtk²] + 2(p + 1)[d − E(0)]

≥ 0, ∀t ∈ (0, ∞).

It follows that

L⁰(t) > L⁰(0) + 2(p + 1)[d − E(0)]t, ∀t ∈ (0, ∞).

This implies that there is t₁ > 0 such that for any t ∈ [t₁, ∞), L⁰(t) > 0.

Consequently, L(t) never vanishes on [t1, ∞). On the other hand, it follows from (3.16)-(3.18) that

L(t)L⁰⁰(t) − (1 +p − 1

4 )L⁰(t)²> 0.

Set N (t) = (L(t))^−p−14 . Then we get N⁰⁰(t) = −p − 1

4 (L(t))^−p+74 [L(t)L⁰⁰(t) − (1 +p − 1

4 )L⁰(t)²] < 0, ∀t ∈ [t1, ∞) as well as N (t1) > 0 and N⁰(t1) < 0. Thus N (t) ≤ N (t1) + (t − t1)N⁰(t1).

So there is a T1∈ (t1, t1+_(p−1)L^4L(t10(t⁾₁)) such that lim

t→T₁⁻

L(t) = ∞,

which contradicts T0= ∞. The theorem is proved.  4. Global existence for E(0) > 0

For the arbitrary initial energy E(0) > 0, using the technique of [4, 24], we derive a sufficient condition on the initial data such that the corresponding local solution of problem (1.1) and (1.2) exists globally.

Lemma 4.1. Let 2 ≤ s < p + 1, φ ∈ H^s∩ H₀¹, ψ ∈ H^s and u(x, t) be the solution of the Cauchy problem (1.1) and (1.2). Assume that the initial data satisfy (1.14) and (1.16). Then, the map

{t → kuk²+ kuxk²+ kuxxk²} is strictly decreasing as long as u(x, t) ∈ Y_T.

Proof. From (3.6), (3.7) and (3.8), we have

L⁰⁰(t) = 2kutk²+ 2kuxtk²+ 2kuxxtk²− 2I(t).

(4.1)

Furthermore, from u(t) ∈ Y_T, we get L⁰⁰(t) < 0 for t ∈ [0, T ). Using (1.16), we obtain

2(φ, ψ) + 2(φx, ψx) + 2(φxx, ψxx) < 0,

which implies L⁰(0) < 0. So it is easy to see that L⁰(t) < L⁰(0) < 0, namely L⁰(t) < 0. Therefore, we get the result of the lemma.  Lemma 4.2. Let 2 ≤ s < p + 1, φ ∈ H^s∩ H₀¹, ψ ∈ H^s and u(x, t) be the weak solution of the Cauchy problem (1.1) and (1.2) with maximal existence time interval [0, T ) such that u ∈ C¹((0, T ), H^s∩ H₀¹) and ut∈ C((0, T ), H^s), here T ≤ +∞. Assume that the initial data satisfy (1.14), (1.15) and (1.16), then u ∈ YT.

Proof. Arguing by contradiction, we suppose that there exists a first time t₂∈ [0, T ) such that

I(u(t₂)) = ku_t(t₂)k²+ ku_xt(t₂)k²+ ku_xxt(t₂)k² (4.2)

and

I(u(t)) > ku_t(t)k²+ ku_xt(t)k²+ ku_xxt(t)k², ∀t ∈ [0, t₂).

From the (3.6)-(3.8) and Lemma 4.1, we obtain that L(t) and L⁰(t) are both strictly decreasing on the interval [0, t2). And by (1.16), for all t ∈ (0, t2), we get

L(u(t)) < kφk²+ kφxk²+ kφxxk²

< −2(φ, ψ) − 2(φ_x, ψ_x) − 2(φ_xx, ψ_xx) −2(p + 1) p + 3 E(0).

Moreover, from the continuity of kuk²+ kuxk²+ kuxxk² in t, we obtain (4.3) L(u(t2)) < −2(φ, ψ) − 2(φx, ψx) − 2(φxx, ψxx) −2(p + 1)

p + 3 E(0).

On the other hand, by Theorem 1.2, (1.10) and (1.11), we get E(0) = E(t2) = 1

2(kut(t2)k²+ kuxt(t2)k²+ kuxxt(t2)k²) + J (u(t2))

= 1

2(ku_t(t₂)k²+ ku_xt(t₂)k²+ ku_xxt(t₂)k²) + p − 1

2(p + 1)(ku_x(t₂)k²+ ku_xx(t₂)k²) + 1

p + 1I(u(t₂)).

Using (4.2), we obtain E(0) = (1

2 + 1

p + 1)(kut(t2)k²+ kuxt(t2)k²+ kuxxt(t2)k²) + (1

2 − 1

p + 1)(kux(t2)k²+ kuxx(t2)k²)

≥ p + 3

2(p + 1)(kut(t2)k²+ kuxt(t2)k²+ kuxxt(t2)k²).

(4.4)

Then from the following equalities

kut(t2)k²= kut(t2) + u(t2)k²− ku(t2)k²− 2(u(t2), ut(t2)), kuxt(t2)k²= kuxt(t2) + ux(t2)k²− kux(t2)k²− 2(ux(t2), uxt(t2)), kuxxt(t2)k²= kuxxt(t2) + uxx(t2)k²− kuxx(t2)k²− 2(uxx(t2), uxxt(t2)) and Lemma 4.1, we get

E(0) ≥ p + 3

2(p + 1)(kut(t2) + u(t2)k²+ kuxt(t2) + ux(t2)k² + kuxxt(t2) + uxx(t2)k²)

− p + 3

2(p + 1)(ku(t2)k²+ kux(t2)k²+ kuxx(t2)k²)

− 2 p + 3

2(p + 1)((u(t2), ut(t2)) + (ux(t2), uxt(t2)) + (uxx(t2), utxx(t2)))

≥ − p + 3

2(p + 1)(ku(t₂)k²+ ku_x(t₂)k²+ ku_xx(t₂)k²)

− 2 p + 3

2(p + 1)((φ(t2), ψ(t2)) + (φx(t2), ψx(t2)) + (φxx(t2), ψxx(t2))).

(4.5) So, (4.6)

L(t₂) ≥ −2((φ(t₂), ψ(t₂))+(φ_x(t₂), ψ_x(t₂))+(φ_xx(t₂), ψ_xx(t₂)))−2(p + 1) p + 3 E(0).

It is obvious that (4.6) contradicts (4.3). This completes the proof.  Proof of Theorem 1.6. From Theorem 1.1, there exists a unique local solution of problem (1.1) and (1.2) defined on a maximal time interval [0, T ), T <

+∞. Let u(t) be the weak solution of problem (1.1) and (1.2) such that u ∈ C¹((0, T ), H¹∩ H₀¹) and u_t∈ C((0, T ), H¹) with (1.14), (1.15) and (1.16).

Then from Lemma 4.2, we have u(x, t) ∈ YT, namely for t ∈ [0, T ), I(u(t)) > ku_t(t)k²+ ku_xt(t)k²+ ku_xxtk².

(4.7)

Therefore from Theorem 1.2, (1.10), (1.11) and (4.7), we obtain E(0) = E(t) = 1

2(kut(t)k²+ kuxt(t)k²+ kuxxt(t)k²) + J (u)

= 1

2(kut(t)k²+ kuxt(t)k²+ kuxxt(t)k²) + p − 1

2(p + 1)(kux(t)k²+ kuxx(t)k²) + 1

p + 1I(u(t))

> p + 3

2(p + 1)(kut(t)k²+ kuxt(t)k²+ kuxxt(t)k²) + p − 1

2(p + 1)(ku_x(t)k²+ ku_xx(t)k²).

From the Poincar´e inequality, that is kvk² ≤ C0kvxk², ∀v ∈ H₀¹, where C0 is a positive constant, it follows that u(x, t) is bounded in C¹((0, T ), H²∩ H₀¹), ut(x, t) is bounded in C¹((0, T ), H²). Hence from Theorem 1.1, it follows that T = ∞ and the solution of problem (1.1) and (1.2) exists globally.  Acknowledgments. The work is partially supported by the NSF-China grant- 11701068, the Fundamental Research Funds for the Central Universities grant- ZYGX2015J096.

References

[1] A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Differential Integral Equations 15 (2002), no. 9, 1061–1072.

[2] G. B. Folland, Real Analysis, second edition, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999.

[3] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equa- tions, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907.

[4] N. Kutev, N. Kolkovska, and M. Dimova, Global existence of Cauchy problem for Boussi- nesq paradigm equation, Comput. Math. Appl. 65 (2013), no. 3, 500–511.

[5] S. Lai and Y. Wu, The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation, Discrete Contin. Dyn. Syst. Ser. B 3 (2003), no. 3, 401–408.

[6] Q. Lin, Y. H. Wu, and R. Loxton, On the Cauchy problem for a generalized Boussinesq equation, J. Math. Anal. Appl. 353 (2009), no. 1, 186–195.

[7] Y. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equa- tions, J. Differential Equations 192 (2003), 109–127.

[8] Y. Liu and R. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D 237 (2008), no. 6, 721–731.

[9] N. Polat and A. Ertas, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl. 349 (2009), no. 1, 10–20.

[10] N. Polat and E. Piskin, Asymptotic behavior of a solution of the Cauchy problem for the generalized damped multidimensional Boussinesq equation, Appl. Math. Lett. 25 (2012), no. 11, 1871–1874.

[11] T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, 3, Walter de Gruyter & Co., Berlin, 1996.

[12] G. Schneider and C. E. Wayne, Kawahara dynamics in dispersive media, Phys. D 152/153 (2001), 384–394.

[13] H. Wang and A. Esfahani, Well-posedness for the Cauchy problem associated to a peri- odic Boussinesq equation, Nonlinear Anal. 89 (2013), 267–275.

[14] , Global rough solutions to the sixth-order Boussinesq equation, Nonlinear Anal.

102 (2014), 97–104.

[15] S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal. 64 (2006), no. 1, 159–173.

[16] S. Wang and G. Xu, The Cauchy problem for the Rosenau equation, Nonlinear Anal.

71 (2009), no. 1-2, 456–466.

[17] S. Wang and H. Xue, Global solution for a generalized Boussinesq equation, Appl. Math.

Comput. 204 (2008), no. 1, 130–136.

[18] H. Taskesen and N. Polat, Existence of global solutions for a multidimensinal Boussinesq type equation with supercritical initial energy, In: First international Conference on Analysis and Applied Mathematics: ICAAM, AIP Conference Proceedings 1470 (2012), 159–162.

[19] H. Taskesen, N. Polat, and A. Erta, On global solutions for the Cauchy problem of a Boussinesq-type equation, Abstr. Appl. Anal. (2012), Art. ID 535031, 10 pp.

[20] Y. Wang and C. Mu, Blow-up and scattering of solution for a generalized Boussinesq equation, Appl. Math. Comput. 188 (2007), no. 2, 1131–1141.

[21] , Global existence and blow-up of the solutions for the multidimensional gener- alized Boussinesq equation, Math. Methods Appl. Sci. 30 (2007), no. 12, 1403–1417.

[22] Y. Wang, C. Mu and J. Deng, Strong instability of solitary-wave solutions for a non- linear Boussinesq equation, Nonlinear Anal. 69 (2008), no. 5-6, 1599–1614.

[23] Y.-Z. Wang and Y.-X. Wang, Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order, Nonlinear Anal. 72 (2010), no. 12, 4500–

4507.

[24] R. Z. Xu, Y. B. Yang, B. W. Liu, J. H. Shen, and S. B. Huang, Global existence and blowup of solutions for the multidimensional sixth-order “good” Boussinesq equation, Z. Angew. Math. Phys. 66 (2015), no. 3, 955–976.

Ying Wang

School of Mathematical Sciences

University of Electronic Science and Technology of China Chengdu 611731, P. R. China

Email address: nadine−1979@163.com